Nonlinear Dynamic Analysis of Macrofiber Composites Laminated Shells

This work presents the nonlinear dynamical analysis of a multilayer d31 piezoelectric macrofiber composite (MFC) laminated shell. The effects of transverse excitations and piezoelectric properties on the dynamic stability of the structure are studied. Firstly, the nonlinear dynamic models of the MFC laminated shell are established. Based on known selected geometrical and material properties of its constituents, the electric field of MFC is presented. The vibration mode-shape functions are obtained according to the boundary conditions, and then the Galerkin method is employed to transform partial differential equations into two nonlinear ordinary differential equations. Next, the effects of the transverse excitations on the nonlinear vibration ofMFC laminated shells are analyzed in numerical simulation andmoderating effects of piezoelectric coefficients on the stability of the system are also presented here. Bifurcation diagram, two-dimensional and three-dimensional phase portraits, waveforms phases, and Poincare diagrams are shown to find different kinds of periodic and chaotic motions ofMFC shells.The results indicate that piezoelectric parameters have strong effects on the vibration control of the MFC laminated shell.


Introduction
MFC is a piezoelectric fiber material, consisting of monotonic piezoelectric material, epoxy matrix, and electrodes with a specific arrangement, which can be considered as homogenized orthotropic materials with arbitrary piezoelectric fiber angles like composite structures.
There are generally three types of piezoelectric material which researches focus on mainly.The first type of the piezoelectric fiber composite material is produced by Smart Material Corp, treated as 1-3 composite.The second type is originally developed by MIT, named active fiber composite actuators.Macrofiber composite (MFC) is referred to as the third type presented by NASA Langley Research Center.Since several applications require conformable and packaged piezoelectric structures or actuators, MFC material become widely applied in both academic and industrial field, which would be used for an easier integration in smart, intelligent, or adaptive structures [1][2][3].Great advantages of the MFC material, which is able to respond to changing environment and controlling structural deformation, have led to a new generation for aerospace structures especially for morphing aircrafts [4].
Theoretical analyses are the basis to obtain dynamical characteristics of MFC material subjected to different excitations.In some researches, piezoelectric characteristics of MFC are studied by nonlinear constitutive equations and numerical simulation, including higher-order terms and corresponding coefficients [5][6][7] or by using strain dependent or effective piezoelectric coefficients in constitutive equations [8][9][10][11].
Recently, different finite element models are developed to analyze the nonlinear dynamical behavior of MFC laminated materials [12][13][14][15][16]. Gohari et al. [17] present an analytical solution to obtain static deformation and optimal shape control of smart laminated cantilever piezoelectric composite hybrid plates and beams under several coupled loads using piezoelectric actuators.Park and Kim [18] show that the aerothermal large deflection of the composite panel can be suppressed using MFC actuators and the excessive actuation of the MFC can cause snap-through phenomena.Kim et al. [19] study the vibration suppression of an end-capped cylindrical shell structure with surface bonded macrofiber composite actuators and analyze the dynamic characteristics of the cylindrical shell structure.
Dano and Jullière [20] investigate the use of macrofiber composite (MFC) actuators to actively control thermally induced deformations in composite structures.Korayem and Homayooni [21] analyze the influence of the applied voltage on vibration frequency of a multilayer piezoelectric microplate as well as the size effects on the macroplate at different boundary conditions.Li et al. [22] present the active control of random vibration for piezoelectric fiber reinforced composites laminated plates and discuss the effect of piezoelectric fiber orientation in the PFRC layers.Suresh Kumar and Ray [23] analyze the geometrically nonlinear vibrations of doubly curved smart sandwich shells integrated with a patch of 1-3 piezoelectric composites active constrained layer damping (ACLD) treatment.
Experimental studies are effective in verifying theoretical results.The effective properties of piezoelectric composites on the interface material are reported by using theoretical and numerical methods in [24,25].Andrianov et al. [26] focus on the evaluation of the homogeneous properties of the active layers in both  31 and  33 MFCs.Biscani et al. [27] study the effective electromechanical properties of MFC transducers and compare the material properties of the system with experimental data.Prasath and Arockiarajan [28,29] present total packing effects on the overall properties of MFC using experimental models.Based on the developed Kirchhoff plate theory, in some articles [30,31], the actuation responses of  33 and  31 MFCs integrated smart structures are studied subjected to transverse excitations.Kashiwao et al. [32] propose an optimization method of a vibration energy harvesting system which is made of MFC piezoelectric elements.

Mechanical Model
MFC mainly consist of piezoceramic fibers, epoxy matrix, and electrodes, which have two different types of structures, named  31 and  33 modes.Here, a  31 MFC middling thick cross-ply laminated shell is considered, where the piezomacrofiber is polarized in the thickness direction and fully embedded in the epoxy matrix as shown in Figure 1.The shell is a cantilever doubly curved cylindrical shell with a rectangular base.The orthogonal curvilinear coordinate is shown in Figure 2, and  and  curves are along the lines of curvatures on the middle surface and  is perpendicular to the middle surface of the shell.Geometric dimensions of the shell in the curvilinear coordinate are the lengths  and , and the thickness ℎ, and the principal radii of the curvatures  1 and  2 .The displacements of an arbitrary point within the shell in the coordinate (, , ) are denoted with , V, and , respectively. is taken as positive vector going outward from the center of the smallest radius of the curvature.The shell is subjected to a transverse excitation given as  =  0 +  1 cos(Ω), as shown in Figure 2.
According to Reddy's third-order theory, the displacement fields at an arbitrary point in the composite shell are given in the following form: where  0 , V 0 , and  0 are the original displacements at the mid-plane of the MFC shell in the , ,  directions;  1 and  2 represent the rotations of transverse normal at the mid-plane about the  and  axes.
Using von Karman's geometric relationship, the strain and displacements of the MFC shell can be written as follows: ) . ( Here,  1 ,  2 are the Lame coefficients of the shell and can be expressed as The stress-strain relationships for the MFC material are given by where   is the electric field intensity and   represents the piezoelectric constant. All piezoelectric fibers are considered to be poled in  (through-thickness) direction.Therefore, it can be assumed that in-plane electric fields vanish (i.e.,  1 =  2 = 0), and the following reduced constitutive equations hold: where , ( = 0, 1, 2, 3, 4, 6) . ( The relationship between curvilinear coordinate system and rectangular coordinate system is  =  1 ,  =  2 ,  = .Substitute these transformations into (1a), (1b), and (1c) and then apply Hamilton's principle; the nonlinear governing equations of motion in terms of generalized displacements for the MFC shell can be obtained as follows: where  1 = 4/3ℎ 2 ,  2 = 3 1 , and  in (6c) is the damping coefficient, and all the parameters of the specific expression can be found in the Appendix.The boundary conditions of the cantilever shell are expressed as , ( = 0, ) .(7e)

Perturbation Analysis
Since vibration amplitudes of the lower frequencies are much larger than that of the higher frequencies for the shell, the mainly dynamical damage or instability of structures is caused by resonances in lower frequencies.Here, the first two modes of the MFC laminated shell are considered.Thus, displacements  0 , V 0 ,  0 ,  1 , and  2 , which satisfy the boundary conditions for the shell, are represented as where cos To obtain the dimensionless equations, the transformation of variables and parameters are introduced as Then, taking all these derived expressions in (8a), (8b), (8c), (8d), and (8e)-(9a), (9b), (9c), (9d), (9e), and (9f) into (6a), (6b), (6c), (6d), and (6e) and applying the Galerkin procedure, a two-degree-of-freedom nonlinear ordinary differential equation of the MFC laminated shell with dimensionless is obtained as follows: Advances in Materials Science and Engineering Here,  14 ,  15 ,  24 , and  25 present the piezoelectric coefficients of the shell.
Considering the case of primary parametric and 1 : 1 internal resonance, the following relations can be established accordingly: where  1 and  2 are the detuning parameters.
To study the oscillations and bifurcations of the established nonlinear system, the method of multiscale is a powerful tool to determine the solutions for conservative and nonconservative systems.Hence, this method is adopted to investigate the nonlinear vibration responses of the MFC shell.
The uniformly approximate solutions of (11a) and (11b) are obtained as where  0 =  and  1 = .
Then, the derivatives with respect to  become Substituting (13a), (13b), (14a), and (14b) into (11a) and (11b) and equating the coefficients of  to zero yield the following differential equations: The averaged equations in the Polar Coordinates form are obtained for the MFC laminated shell as follows:

Numerical Simulation
In this section, a series of numerical experiments are conducted for the nonlinear dynamical behavior of the MFC laminated shell.The nonlinear governing equations (17a), (17b), (17c), and (17d) are used to perform the numerical simulation through the Runge-Kutta algorithm.Whereas the transverse excitation is one of the most important factors on the nonlinear vibration of the system, the excitation  1 is chosen as a controlling parameter in the subsequent studies.
The bifurcation diagrams of Poincare sections for the displacements of the middle surface of the shell are shown in Figure 4 with the transverse excitation increasing from 700 to 2000.The complex nonlinear vibration responses of the system are discussed corresponding to the above parameters.The system keeps the chaotic motion until  1 increased to 800, at which point period motions of the system occur.After short windows of period- motions occur in 950-1000, the motions of the system dominantly enter into chaotic motions thereafter.Then the chaotic motion is observed again until the transverse excitation  1 equals 1500.The motion status of the MFC laminated shell alternately appears from unstable to stable and then again to unstable, which depends on the amplitude of the transverse excitation  1 .
To reveal the specific form of different sections in the bifurcation diagram, the response-frequency curves, phase portraits, power spectrums, and waveforms of the shell are depicted as shown in Figures 5(a)-5(f).Diagrams (a) and (c) are, respectively, the wave forms on the planes (,  1 ) and (,  3 ), and Diagrams (b) and (d) are the two-dimensional phase portraits on the planes ( 1 ,  2 ) and ( 3 ,  4 ).Diagrams (e) show the three-dimensional phase portraits in the space ( 1 ,  2 ,  3 ) while Diagrams (f) represent the Poincare diagram.The typical chaotic motion of the system is given as in Figure 5 when the amplitude of the transverse excitation arrives at 760.
From the frequency-response curves of the system analyzed, it is known that the structure stiffness would alter when the piezoelectric coefficients of the system changed, so it would be effective to adjust the motion status of the MFC shell from unstable to stable through modulating the piezoelectric coefficients of the system.Fix the system parameters, based on which Figure 5 is depicted, and only adjust the piezoelectric parameters  25 to perform numerical simulations and the results are shown in Figure 6.The bifurcation diagram reveals that the system would enter into periodic motions when  25 is increased from 0 to 1.6.The specific periodic motion form of the system is also shown in Figure 7 when  25 is equal to 2.
In order to describe the influence of the piezoelectric parameters on the nonlinear vibrations of the system profoundly, another set of parameters is selected in the following:  = 0.6,   8 that the response of the system varies from periodic motions to chaotic motions when the transversal excitation increases from 3 to 18.The different periodic and chaotic motions of the system are also given as in Figures 9-10, respectively, when  1 equals 5 and 12.
Based on the motion of the system in Figure 8, we only increase the piezoelectric parameter  25 to describe the vibration responses of the system.The bifurcation diagram of the system also shows the evolution from chaos to period- and then back into chaos again as shown in Figure 11.It is also found that the system enters into periodic motion from chaotic motion when the piezoelectric parameter  25  increases from 1.0 to 2.0. Figure 12 shows the specific periodic motion of the system as  25 equals 1.5.
From the results of the numerical simulation, it is discovered that the piezoelectric parameters could adjust the vibration responses of the structure effectively.Since the piezoelectric parameters could vary the stiffness of the shell, it is generalized: once the inherent frequency of the system changes, the resonance would be restrained for piezoelectric structures.

Conclusion
In this paper, nonlinear dynamical behaviors of a simply supported cantilever MFC shell are presented, which is subjected to transverse loads.Based on known geometrical and material properties of its constituents, their electric field dependence is presented.The vibration mode-shape functions are obtained according to the boundary conditions, and then Galerkin method is employed to transform the partial differential equations into two nonlinear ordinary differential equations.The illustrative case of 1 : 1 internal resonances is considered.The externally excited system is transformed into a set of averaged equations by using the method of multiple scales.Next, the effect of the transverse excitations and the piezoelectric coefficients on the MFC laminated shell is described in numerical simulation.
The results of the numerical simulation demonstrate the complex nonlinear vibration responses of the MFC shell that occurred under the transverse excitation, including the periodic and chaotic motions.The energy transformation existed between two resonances modes.The appropriate control technique of the forcing excitations contributes significantly to the responses of autonomous nonlinear systems.It is also revealed that the piezoelectric parameters of the structure could adjust the dynamic stability of the structure from unstable to stable which would be a good way to control vibration responses for MFC structures.

Figure 5 :
Figure 5: The chaotic motion of the shell.